arXiv subjects

Zoé Agathe-Nerine

Publications and source records attributed to Zoé Agathe-Nerine.

3 recordsLinked to original sources

arXiv2023

Stability of wandering bumps for Hawkes processes interacting on the circle

We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the circle $[-π, π]$, and the connectivity between neurons is given by a cosine kernel. The firing rate function is a sigmoid. The large population limit admits a locally stable manifold of stationary solutions. The main result of the paper concerns the long-time proximity of the synaptic voltage of the population to this manifold in polynomial times in $N$. We show in particular that the phase of the voltage along this manifold converges towards a Brownian motion on a time scale of order $N$.

math.PR↗

arXiv2022

Long-term stability of interacting Hawkes processes on random graphs

We consider a population of Hawkes processes modeling the activity of $N$ interacting neurons. The neurons are regularly positioned on the segment $[0,1]$, and the connectivity between neurons is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices through a spatial kernel. The main result of the paper concerns the longtime stability of the synaptic current of the population, as $N\to\infty$, in the subcritical regime in case the synaptic memory kernel is exponential, up to time horizons that are polynomial in $N$.

math.PR↗

arXiv2022

Multivariate Hawkes processes on inhomogeneous random graphs

We consider a population of $N$ interacting neurons, represented by a multivariate Hawkes process: the firing rate of each neuron depends on the history of the connected neurons. Contrary to the mean-field framework where the interaction occurs on the complete graph, the connectivity between particles is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices. We address the well-posedness of this system and Law of Large Numbers results as $N\to\infty$. A crucial issue will be to understand how spatial inhomogeneity influences the large time behavior of the system.

math.PR↗